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Fractional diffusion with two time scales

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  • Baeumer, B.
  • Meerschaert, M.M.

Abstract

Moving particles that rest along their trajectory lead to time-fractional diffusion equations for the scaling limit distributions. For power law waiting times with infinite mean, the equation contains a fractional time derivative of order between 0 and 1. For finite mean waiting times, the most revealing approach is to employ two time scales, one for the mean and another for deviations from the mean. For finite mean power law waiting times, the resulting equation contains a first derivative as well as a derivative of order between 1 and 2. Finite variance waiting times lead to a second-order partial differential equation in time. In this article we investigate the various solutions with regard to moment growth and scaling properties, and show that even infinite mean waiting times do not necessarily induce subdiffusion, but can lead to super-diffusion if the jump distribution has non-zero mean.

Suggested Citation

  • Baeumer, B. & Meerschaert, M.M., 2007. "Fractional diffusion with two time scales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 237-251.
    • Handle: RePEc:eee:phsmap:v:373:y:2007:i:c:p:237-251
    DOI: 10.1016/j.physa.2006.06.014
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    References listed on IDEAS

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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    3. Baeumer, B. & Benson, D.A. & Meerschaert, M.M., 2005. "Advection and dispersion in time and space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 245-262.
    4. Bondesson, Lennart & Kristiansen, Gundorph K. & Steutel, Fred W., 1996. "Infinite divisibility of random variables and their integer parts," Statistics & Probability Letters, Elsevier, vol. 28(3), pages 271-278, July.
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    Cited by:

    1. Mijena, Jebessa B. & Nane, Erkan, 2014. "Correlation structure of time-changed Pearson diffusions," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 68-77.
    2. Ali Balcı, Mehmet, 2017. "Time fractional capital-induced labor migration model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 91-98.

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